Cosmos Automaton: A Deterministic Fractal Automaton Generating Primes

This paper introduces the "Cosmos Automaton" (CA), a deterministic system that generates prime numbers through iterative morphisms rather than arithmetic primality testing. Formalized as a non-stationary S-adic substitution system, the CA generates its own directive sequence endogenously through an internal feedback loop. Analysis of the incidence matrix Mp reveals a recursive growth factor of p-2 for twin prime templates, matching OEIS A059861 and the combinatorial core of the Hardy-Littlewood k-tuple conjecture. The model defines a linear Stability Zone that ensures pointwise convergence to a unique aperiodic limit-word, forming a Cantor-like set of measure zero termed "Heeren Dust." This framework provides a rigorous algorithmic bridge between automata theory and the structural emergence of primes.